Six people who all know each other are in a room. Every two people either like each other or dislike each other. Show that there is a group of three people who either all like each other or all dislike each other.

We put our six people on a circle. We connect people who like each other with chords. In the figure, *A* likes everybody. Now it’s clear that if **any two** among *B, C, D, E,* and *F* like each other, then we’ll have a triangle, indicating the set of three who like each other. We added dotted lines to show, then, that a dotted-line triangle is inevitable.

Suppose *A* is friends with everybody except *B*. You can work out what happens. Here in the second figure, *A* is friends with only *D, E,* and *F*. If any of *D, E,* and *F* are friends we have a triangle. So then they must describe each other — and we still have a (dotted) triangle.

All possibilities lead to some triangle, solid or dotted.